ASYMPTOTIC EXPANSION OF DOUBLE LAPLACE-TYPE INTEGRALS WITH A CURVE OF MINIMAL POINTS AND APPLICATION TO AN EXIT TIME PROBLEM

In this paper we consider the problem of the asymptotic expansion of double Laplace-type integrals, in the case when the set gamma of points where the phase achieves its absolute minimum is a simple curve. It will be shown that the asymptotic behaviour of such integrals is governed by the order of degeneracy of normal derivatives of the phase with respect to the curve gamma. Complete asymptotic expansions will be constructed if that order is constant along gamma, and the first two coefficients will be explicitly computed. If not, a uniform asymptotic expansion method, involving special functions, is suggested. (C) 2017 Mathematical Institute Slovak Academy of Sciences